We present a spectral approach for detecting and analyzing rotational and reflectional symmetries in n-dimensions. Our main contribution is the derivation of a symmetry detection and analysis scheme for sets of points in IRn and its extension to image analysis by way of local features. Each object is represented by a set of points S 2 IRn, where the symmetry is manifested by the multiple self-alignments of S. The alignment problem is formulated as a quadratic binary optimization problem, with an efficient solution via spectral relaxation. For symmetric objects, this results in a multiplicity of eigenvalues whose corresponding eigenvectors allow the detection and analysis of both types of symmetry. We improve the scheme's robustness by incorporating geometrical constraints into the spectral analysis. Our approach is experimentally verified by applying it to 2D and 3D synthetic objects as well as real images.

We present a computational approach to high-order matching of data sets in IR(d). Those are matchings based on data affinity measures that score the matching of more than two pairs of points at a time. High-order affinities are represented by tensors and the matching is then given by a rank-one approximation of the affinity tensor and a corresponding discretization. Our approach is rigorously justified by extending Zass and Shashua's hypergraph matching to high-order spectral matching. This paves the way for a computationally efficient dual-marginalization spectral matching scheme. We also show that, based on the spectral properties of random matrices, affinity tensors can be randomly sparsified while retaining the matching accuracy. Our contributions are experimentally validated by applying them to synthetic as well as real data sets.